Algebraic properties

Prime powers are powers of prime numbers. Every prime power (except powers of 2) has a primitive root; thus the multiplicative group of integers modulo pⁿ (that is, the group of units of the ring Z/pⁿZ) is cyclic.^[1]

The number of elements of a finite field is always a prime power and conversely, every prime power occurs as the number of elements in some finite field (which is unique up to isomorphism).^[2]

Combinatorial properties

A property of prime powers used frequently in analytic number theory is that the set of prime powers which are not prime is a small set in the sense that the infinite sum of their reciprocals converges, although the primes are a large set.^[3]

Divisibility properties

The totient function (φ) and sigma functions (σ₀) and (σ₁) of a prime power are calculated by the formulas

${\displaystyle \varphi (p^{n})=p^{n-1}\varphi (p)=p^{n-1}(p-1)=p^{n}-p^{n-1}=p^{n}\left(1-{\frac {1}{p}}\right),}$

${\displaystyle \sigma _{0}(p^{n})=\sum _{j=0}^{n}p^{0\cdot j}=\sum _{j=0}^{n}1=n+1,}$

${\displaystyle \sigma _{1}(p^{n})=\sum _{j=0}^{n}p^{1\cdot j}=\sum _{j=0}^{n}p^{j}={\frac {p^{n+1}-1}{p-1}}.}$

All prime powers are deficient numbers. A prime power pⁿ is an n-almost prime. It is not known whether a prime power pⁿ can be a member of an amicable pair. If there is such a number, then pⁿ must be greater than 10¹⁵⁰⁰ and n must be greater than 1400.